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A symmetric solution of a multipoint boundary value problem at resonance. (English) Zbl 1137.34313
Summary: We apply a coincidence degree theorem of Mawhin to show the existence of at least one symmetric solution of the nonlinear second-order multipoint boundary value problem
\begin{alignedat}{2} 2 u''(t)&= f(t,u(t),|u'(t)|), &\quad&t \in (0,1),\\ u(0)&= \sum_{i=1}^n \mu_i u(\xi_i),\;u(1-t) = u(t), &\quad &t \in [0,1], \end{alignedat}
where $$0 < \xi_1 <\xi_2 < \cdots < \xi_n \leq {1}/{2}$$, $$\sum_{i=1}^n \mu_i = 1$$, $$f: [0,1]\times \mathbb{R}^{2} \rightarrow \mathbb{R}$$ with $$f(t,x,y) = f(1-t,x,y)$$, $$(t,x,y) \in [0,1]\times \mathbb{R}^{2}$$, satisfying the Carathéodory conditions.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
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